Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann Problem
In this brief note, we study the problem of asymptotic behavior of the solutions for non-resonant, singularly perturbed linear Neumann boundary value problems <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow&g...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-08-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/11/8/394 |
| _version_ | 1797432515188752384 |
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| author | Robert Vrabel |
| author_facet | Robert Vrabel |
| author_sort | Robert Vrabel |
| collection | DOAJ |
| description | In this brief note, we study the problem of asymptotic behavior of the solutions for non-resonant, singularly perturbed linear Neumann boundary value problems <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><msup><mi>y</mi><mrow><mo>″</mo></mrow></msup><mo>+</mo><mi>k</mi><mi>y</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>y</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>y</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>b</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> with an indication of possible extension to more complex cases. Our approach is based on the analysis of an integral equation associated with this problem. |
| first_indexed | 2024-03-09T10:01:42Z |
| format | Article |
| id | doaj.art-9aa30afd1aeb4b62bef27d22e018707e |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T10:01:42Z |
| publishDate | 2022-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-9aa30afd1aeb4b62bef27d22e018707e2023-12-01T23:24:37ZengMDPI AGAxioms2075-16802022-08-0111839410.3390/axioms11080394Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann ProblemRobert Vrabel0Institute of Applied Informatics, Automation and Mechatronics, Slovak University of Technology in Bratislava, Bottova 25, 917 01 Trnava, SlovakiaIn this brief note, we study the problem of asymptotic behavior of the solutions for non-resonant, singularly perturbed linear Neumann boundary value problems <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><msup><mi>y</mi><mrow><mo>″</mo></mrow></msup><mo>+</mo><mi>k</mi><mi>y</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>y</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>y</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>b</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> with an indication of possible extension to more complex cases. Our approach is based on the analysis of an integral equation associated with this problem.https://www.mdpi.com/2075-1680/11/8/394singular perturbationlinear ordinary differential equationNeumann boundary value problem |
| spellingShingle | Robert Vrabel Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann Problem Axioms singular perturbation linear ordinary differential equation Neumann boundary value problem |
| title | Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann Problem |
| title_full | Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann Problem |
| title_fullStr | Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann Problem |
| title_full_unstemmed | Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann Problem |
| title_short | Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann Problem |
| title_sort | non resonant non hyperbolic singularly perturbed neumann problem |
| topic | singular perturbation linear ordinary differential equation Neumann boundary value problem |
| url | https://www.mdpi.com/2075-1680/11/8/394 |
| work_keys_str_mv | AT robertvrabel nonresonantnonhyperbolicsingularlyperturbedneumannproblem |